Let \(\mu ^+\)  be \(\sigma \) -invariant on \(\Sigma _a^+\) . Let \(\zeta \)  be a \(\mu ^+\) -measurable partition generates a \(\sigma \) -algebra \(\widehat{\zeta }\)  that is equal to the \(\sigma \) -algebra \(\mathcal {I}_a=\lbrace B\in \mathcal {B}(\Sigma _a):\sigma ^{-1}(B)=B\rbrace \)  modulo sets of zero \(\mu \) -measure. Then the conditional measures \(\mu ^\zeta _{\mathbf {i}^+}\)  for \(\mu ^+\) -a.e. \(\mathbf {i}^+\in \Sigma _a^+\)  form an ergodic decomposition of \(\mu \) . We have the following lemma which can be deduced from [1]}.
